MEN OF MATHEMATICS denumerable infinity, as first stated: the substitutions can be counted off 1,2,3, ... , and are not as numerous as the points '' on a line. Poincare actually constructed such functions and developed their most important properties in a series of papers in the 1880*8. Such functions are.called automorphic. Only two remarks need be made here to indicate what Poin- care achieved by this wonderful creation. First, his theory includes that of the elliptic functions as a detail. Second, as the distinguished French mathematician Georges Humbert said, Poincare found two memorable propositions which 'gave him the keys of the algebraic cosmos': Two automorphic functions* invariant under the same group are connected by an algebraic equation; Conversely, the co-ordinates of a point on any algebraic curve can be expressed in terms of automorphic functions, and hence by uniform functions of a single parameter (variable). An algebraic curve is one whose equation is of the type P(x,y) = 0, where P(x,y) is a polynomial in x and y. As a simple example, the equation of the circle whose centre is at the origin - (0,0) - and whose radius is 0, is #2 + y2 = a2. According to the second of Poincare's *keys% it must be possible to express a?,y as automorphic functions of a single parameter, say t. It is; for if x = a cos t and y == a sin £, then, squaring and adding, we get rid of t (since cos21 + sin21 = 1), and find #2 + yz = a2. But the trigonometric functions cos t, sin t are special cases of elliptic functions, which in turn are special cases of automorphic functions. The creation of this vast theory of automorphic functions was but one of many astonishing things in analysis which Poinear6 did before he was thirty. Nor was all his time devoted to analy- sis; the theory of numbers, parts of algebra, and mathematical astronomy also shared Ms attention. In the first he recast the Gaussian theory of binary quadratic forms (see chapter on * PoincarS called some of his functions 'Fuchsian*, after the German mathematician Lazarus Fuchs (1833-1902) one of the creators of the modern theory of differential equations, for reasons that need not be gone into here. Others he called 'Kleinian' after Felix Klein - in ironic acknowledgement of disputed priority. 596